%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hurwitz spaces of $\PP^1$ covers}\label{sec:hproj}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we recall the definition of the moduli
space of admissible covers of~\cite{harrismumford} as a compactification
of the classical Hurwitz space (see also \cite{harrismorrison}),
and prove formulas to compute recursively intersection numbers
of $\psi$-classes on these moduli spaces. These intersection numbers
will appear as coefficients and multiplicities in the volume
recursion. Along the way we introduce basic notions on stable graphs and
level functions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hurwitz spaces and admissible covers}
\label{sec:genHurwitz}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $d,g,$ and~$g'$ be non-negative integers.
Let $\Hmu = (\mu^{(1)}, \cdots, \mu^{(n)})$ 
be a {\em ramification profile} consisting of~$n$ partitions.
We define the {\em Hurwitz space} $H_{d,g,g'}(\Hmu)$ to be the moduli space 
parametrizing branched covers of smooth connected curves $p\colon X \to Y$
of degree~$d$ with  profile $\Hmu$ and such that the genera of~$X$ and~$Y$
are given by~$g$ and~$g'$ respectively. That is, $p$ is ramified over~$n$
points and over the $i$-th branch point the sheets coming together form 
the partition $\mu^{(i)}$ (completed by singletons if $|\mu^{(i)}| < \deg(p)$).
\par
The Hurwitz space $H_{d,g,g'}(\Hmu)$ has a natural compactification
$\BH_{d,g,g'}(\Hmu)$ parame\-trizing {\em admissible covers}. An admissible 
cover $p\colon X\to Y$ is a finite morphism of connected nodal curves such that
\begin{itemize}
\item[i)] the smooth locus of~$X$ maps to the smooth locus of~$Y$ and the nodes
of~$X$ map to the nodes of $Y$,
\item[ii)] at each node of $X$ the two branches have the same ramification order, and
\item[iii)] the target curve~$Y$ marked with the branch points is stable.
\end{itemize}

The space $\BH_{d,g,g'}(\Hmu)$ is equipped with two forgetful maps
$$
\xymatrix{
& \BH_{d,g,g'}(\Hmu) \ar[ld]_{f_S} \ar[rd]^{f_T} &\\
\oM_{g,m}&& \oM_{g',n}
}
$$
obtained by mapping an admissible cover to the stabilization of the source or 
the target. Here~$n$ denotes the number of branch points or equivalently
the length of~$\Hmu$ and~$m$ denotes the number of ramification points
or equivalently the number of parts (of length~$>1$) of all the $\mu^{(i)}$.  
The {\em Hurwitz number} $N^{\circ}_{d,g,g'}(\Hmu)$ is the degree of 
the map~$f_T$, or equivalently the number of connected covers $p\colon X \to Y$
of degree~$d$ with profile~$\Hmu$ and the location of the
branch points fixed in $Y$. We also denote by $N_{d,g,g'}(\Hmu)$ the Hurwitz 
number of covers without requiring~$X$ to be connected. We remark that each 
cover is counted with weight given by  the reciprocal of the order of its 
automorphism group, as is standard for the Hurwitz counting problem.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Intersection of $\psi$-classes on Hurwitz spaces} \label{subsec:n=2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From now on in this section we will consider the special case $g'=0$.
Let $\mu [0]=(m_1,\ldots,m_n)$  be a list of non-negative integers
and $\mu [\infty ]=(p_1,\ldots,p_k)$ a list of positive integers.
We consider the Hurwitz space with profile $\Hmu$ given
by $\mu^{(i)} = (m_i+1)$ for $i \leq n$ and $\mu^{(n+1)} = (p_1,\ldots,p_k)$
such that $d=\sum_{i=1}^k p_i$, i.e.\ we consider
$$\BH_{\proj^1}(\mu [0],\mu [\infty ]) \=
\BH_{d,g,0}((m_1+1),\ldots,(m_n+1),(p_1,\ldots,p_k))\,.
$$ 
By the Riemann-Hurwitz formula, the genus $g$ of the covering surfaces satisfies that 
$$ 2-2g = k+d - \sum_{i=1}^n m_i = k + \sum_{i=1}^k p_i -  \sum_{i=1}^n m_i\,.$$ 
The forgetful map $f_S$ goes from $\BH_{\proj^1}(\mu [0],\mu [\infty ])$ to $\oM_{g,n+k}$, where we assume that the first $n$ marked points are the first $n$ ramification points and the preimages of the last branch point are the $k$ last marked points. Since there are $n+1$ branch points in the target surface of genus zero, we conclude that 
$\dim \BH_{\proj^1}(\mu [0],\mu [\infty ]) = \dim \barmoduli[0,n+1] = n-2$.  
\par
For $g = 0$, define the following intersection numbers on the Hurwitz spaces 
\be \label{eq:HNdef}
  h_{\proj^1}(\mu [0],\mu [\infty ]) \=
\int_{\BH_{\proj^1}(\mu [0 ],\mu [\infty ])} f_S^*\left(\prod_{i=3}^{n} \psi_i\right)\,.
\ee 
The definition of $\psi$-classes will be recalled in Section~\ref{sec:RelMZInt}. 
If $n=2$, then $\BH_{\proj^1}(\mu [0 ],\mu [\infty ])$ is of dimension zero, and hence the intersection number on the right is just the number
of points of the Hurwitz space, i.e.\  
$$h_{\proj^1}((m_1,m_2),\mu [\infty ])
\= N^{\circ}_{d,0,0}((m_1+1), (m_2+1), (p_1,\ldots,p_k))\,.$$
Again we emphasize that the Hurwitz number on the right-hand side is counted with
weight $1/|{\rm Aut}|$ for each cover. Correspondingly
the intersection numbers are computed on the Hurwitz space treated as a stack. Our goal for the rest of the section is to show the following result.  
\par
\begin{Prop} \label{prop:intpsiPP1}
For $n = 2$, $h_{\proj^1}((m_1,m_2),\mu [\infty ])$ can be computed by the coefficient extraction
\bes
h_{\proj^1}((m_1,m_2),\mu [\infty ])
\= (k-1)! [t^{m_1+1}]  \prod_{i=1}^k \frac{\,\, t-t^{p_i+1}}{\!\!\! 1-t }\,,
\ees
and for $n \geq 3$, $h_{\proj^1}(\mu [0],\mu [\infty ])$ can be computed recursively by the sum
\bes
h_{\proj^1}(\mu[0],\mu[\infty])
\=\sum_{\Gamma \in {\rm RT}(\mu[0],\mu[\infty])_{1,2}} h(\Gamma)\, 
\ees
over rooted trees.
\end{Prop}
\par
The definitions of rooted trees and the local contributions $h(\Gamma)$
are given in Sections~\ref{subsec:RT} and~\ref{sec:HurRT} respectively.
The above formula for the Hurwitz number obviously agrees
with~\eqref{eq:defhP1}.
\par
\begin{proof}[Proof of Proposition~\ref{prop:intpsiPP1}, case $n=2$]
Let $S_d$ be the symmetric group acting on $[\![1,d]\!] = \{1,\ldots, d \}$, 
where $d = p_1 + \cdots + p_k = m_1 + m_2 + 2 - k$. Define the set of Hurwitz 
tuples 
$$A(m_1,m_2,\mu [\infty ]) \= 
\{ (\sigma_1,\sigma_2,\sigma_\infty) \} \subset S_d \times S_d \times S_d$$ 
such that 
\begin{itemize}
\item the permutation $\sigma_\infty$ is in the conjugacy class
of $(p_1, \ldots,p_k)$ and we fix a bijection of its cycles with $[\![1,k]\!]$
such that the $i$-th cycle has length $p_i$,
\item the partitions $\sigma_1$ and $\sigma_2$ are cycles of order
$m_1+1$ and $m_2+1$ respectively,
\item the relation $\sigma_1\circ \sigma_2=\sigma_\infty$ holds, and  
\item the group generated by $\sigma_1$, $\sigma_2$ and $\sigma_\infty$ acts transitively on $[\![1,d]\!]$.
\end{itemize}
Then the (weighted) Hurwitz number $h_{\proj^1}((m_1,m_2),\mu [\infty ]) = |A(m_1,m_2,\mu [\infty ])| / d!$.  
\par
The second and third conditions above imply that the union of the supports of the cycles $\sigma_1$ and $\sigma_2$ is $[\![1,d]\!]$. Therefore, $\sigma_1$ and $\sigma_2$ contain exactly 
$(m_1+1)+(m_2+1)-d=k$ common elements. We can write  
$$ \sigma_1 = (a_1,\ldots, a_{i_1-1}, c_1; a_{i_1+1}, \ldots, a_{i_2-1}, c_2; \ldots ; a_{i_{k-1}+1}, \ldots, a_{i_k-1}, c_k)\,, $$
where $1\leq i_1 < i_2 < \cdots < i_k = m_1+1$ and $c_1, \ldots, c_k$ are the common elements of $\sigma_1$ and $\sigma_2$. Since $\sigma_1\circ \sigma_2=\sigma_\infty$ is of conjugacy type $(p_1,\ldots,p_k)$, it is easy to see that $\sigma_2$ must be of the form
$$ \sigma_2 \= (b_1,\ldots, b_{j_1-1}, c_k; b_{j_1+1}, \ldots, b_{j_2-1}, c_{k-1}; \ldots ; b_{j_{k-1}+1}, \ldots, b_{j_k-1}, c_1)\,, $$
for certain $b_i$, such that 
$$ \{ j_1+i_1, (j_2 - j_1) + (i_{k} - i_{k-1}), \ldots, (j_{k} - j_{k-1}) + (i_2 - i_1) \} = \{p_1 + 1, p_2 + 1, \ldots, p_k + 1\}\,. $$ 
If $\tau$ is a permutation on $[\![1,k]\!]$ such that $(j_{k+1-\ell} - j_{k-\ell}) + (i_{\ell+1} - i_\ell) = p_{\tau(\ell)} + 1$, then we have $1\leq i_{\ell+1} - i_\ell \leq p_{\tau(\ell)}$. Conversely, such $\tau$ and $i$-indices determine the $j$-indices. 
\par 
There are $m_1! { d \choose m_1+1}$ choices for $\sigma_1$. Fixing $\sigma_1$, 
to construct $\sigma_2$ we first choose 
\begin{itemize}
\item a permutation $\tau \in S_k$, and then 
\item a partition $(i_1, i_2 - i_1, \ldots, i_k - i_{k-1})$ of $m_1+1$ 
such that $1\leq i_{\ell+1} - i_\ell \leq p_{\tau(\ell)}$ for all $\ell$.
\end{itemize}
This gives $k! [t^{m_1+1}]\prod_{i=1}^k (t+\cdots+t^{p_i})$ choices. Choose $c_1$ out of the elements in $\sigma_1$, which gives $m_1+1$ choices. Along with the $i$-indices this determines the elements $c_2, \ldots, c_k$ as well as the set of $b$'s as the complement of the union of $a$'s and $c$'s. Finally $\sigma_2$ is determined by arranging the $b$'s, which gives $(m_2+1-k)!$ choices. Note that in this process only 
the cyclic order of $(c_1, \ldots, c_k)$ matters and we cannot actually determine which one is the first $c$, hence we need to divide the final count by $k$.   
\par 
In summary, we conclude that 
$$ |A(m_1,m_2,\mu [\infty ])|  =  (m_1+1)!(m_2+1-k)! { d \choose m_1+1} \cdot (k-1)! [t^{m_1+1}]\prod_{i=1}^k (t+\cdots+t^{p_i})\,. $$
Since $d = m_1+m_2 + 2 - k$, we obtain
$$h_{\proj^1}((m_1,m_2),\mu [\infty ]) 
\= |A(m_1,m_2,\mu [\infty ])| / d! 
\= (k-1)! [t^{m_1+1}]\prod_{i=1}^k \frac{t(1-t^{p_i})}{1-t} $$
using that $(m_1+1)!\,(m_2+1-k)!\, { d \choose m_1+1} = d!$. 
\end{proof}
\par
We remark that the above Hurwitz counting problem can also be interpreted by the angular data of the configurations of saddle connections joining two zeros $z_1$ and $z_2$ of order $m_1$ and $m_2$ respectively in the setting of \cite{emz}.  
Suppose $f\colon \proj^1 \to \proj^1$ is a branched cover parameterized in the Hurwitz space $H_{d,0,0}((m_1+1), (m_2+1), (p_1,\ldots,p_k))$, where we treat $f$ as a meromorphic function with $k$ poles of order $p_1, \ldots, p_k$. Then the meromorphic differential $\eta = df$ has two zeros of order $m_1$ and $m_2$ as well as $k$ poles of order $p_1+1, \ldots, p_k+1$ with no residue. Conversely given such $\eta$, integrating $\eta$ gives rise to a desired branched cover $f$. Such $\eta$ can be constructed using flat geometry as in \cite[Section 2.4]{chenPB}. In particular, it is determined by the angles $2\pi (a'_i+1)$ between the saddle connections (clockwise) at $z_1$ and the angles $2\pi (a''_i+1)$ (counterclockwise) at $z_2$, such that 
$ \sum_{i=1}^k (a'_i+1) = m_1 + 1$, $\sum_{i=1}^k (a''_i+1) = m_2 + 1$ and $a'_i + a''_{i} + 2= p_i + 1$. We see again that the choices involve a partition $(a'_1+1, \ldots, a'_k+1)$ 
 of $m_1 + 1$ such that $a'_i + 1 \leq p_i$ for all $i$.   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Level graphs and rooted trees}\label{subsec:RT}
%%%%%%%%%%%%%%%%%%%%%

The boundary of the Deligne-Mumford compactification $\barmoduli[g,n]$ 
is naturally stratified by the topological types of the stable marked 
surfaces. These boundary strata  are in one-to-one correspondence with 
stable graphs, whose definition we recall below. The boundary strata of 
Hurwitz spaces and of moduli spaces of Abelian differentials are encoded 
by adding level structures and twists to stable graphs.
\par
\begin{Defi}\label{def:stgraph} A {\em stable graph} is the datum of
\begin{equation*}
\Gamma \= (V, H, g\colon V \to \mathbb{N}, a\colon H \to V, i\colon H \to H, E, L\simeq [\![1,n]\!])
\end{equation*}
satisfying the following properties:
\begin{itemize}
\item $V$ is a vertex set with a genus function $g$;
\item $H$ is a half-edge set equipped with a vertex assignment $a$
and an involution~$i$ (and we let $n(v) = |a^{-1}(v)|$);
\item $E$, the edge set, is defined as the set of length-$2$ orbits of~$i$
in $H$ (self-edges at vertices are permitted);
\item $(V, E)$ define a connected graph;
\item $L$ is the set of fixed points of $i$, called {\em legs} or {\em markings}, and is identified with $[\![1,n]\!]$;
\item for each vertex $v$, the stability condition $2g(v) - 2 + n(v) > 0$
holds.
\end{itemize}
Let $v(\Gamma)$ and $e(\Gamma)$ denote the cardinalities of $V$ and $E$ respectively.  The genus of $\Gamma$ is defined 
by $\sum_{v\in V(\Gamma)} g(v) + e(\Gamma) - v(\Gamma) + 1$. 
\end{Defi}
\par
We denote by ${\rm Stab}(g,n)$ the set of stable graphs of genus~$g$
and with $n$~legs. A stable graph is said of {\em compact type} if
$h^1(\Gamma)=0$, i.e.\ if the graph has no loops, which is thus a {\em tree}.  
\par
\medskip
We will use two extra structures on stable graphs, called level functions
and twists. As in \cite{strata} we define a {\em level graph} to be a
stable graph $\Gamma$
together with a level function $\ell\colon V(\Gamma) \to \RR_{\leq 0}$. An edge 
with the same starting and ending level is called a {\em horizontal edge}. 
A {\em bi-colored graph} is a level graph with two levels (in which case we 
normalize the level function to take values in $\{0,-1\}$) that has 
no horizontal edges. We denote the set of bi-colored graphs by ${\rm Bic}(g,n)$.
\par
Recall the notation $\mu[0] = (m_1, \ldots, m_n)$ and $\mu[\infty] = (p_1, \ldots, p_k)$
where $m_i \geq 0$ and $p_j > 0$ for all $i$ and $j$.    
\par
\begin{Defi} \label{def:twist}
Let $\Gamma$ be a stable graph in ${\rm Stab}(g,n+k)$.
A {\em twist assignment} on $\Gamma$ of type $(\mu[0],\mu[\infty])$
is a function
$\bfp\colon H(\Gamma)\to \ZZ$ satisfying the following conditions: 
\begin{itemize}
\item If $(h,h')$ is an edge, then $\bfp(h)+\bfp(h')=0$.
\item For all $1\leq i\leq n$, the twist of the $i$-th leg is $m_i+1$ and
for all $1\leq i\leq k$ the twist of the $(n+i)$-th leg is $-p_i$. 
\item For all vertices $v$ of $\Gamma$ \bes
2g(v) -2 + n(v) \= \!\!\!\!\! \sum_{h\in L, \, a(h) = v} \bfp(h)\,.
\ees
\end{itemize}
\end{Defi}
\par
Suppose the graph~$\Gamma$ comes with a level structure~$\ell$. We say that
a twist~$p$ is {\em compatible} with the level structure if for all
edges $(h,h')$ the condition $\bfp(h)>0$ implies that $\ell(a(h))>\ell(a(h'))$, 
and respectively for the cases~$<$ and~$=$. In this case
we call the triple $(\Gamma,\ell,\bfp)$ a {\em twisted level graph}. For the reader
familiar with related results of compactifications of strata of Abelian differentials, 
the above definition characterizes  twisted differentials (or canonical divisors)
in \cite{strata} and \cite{fapa} (regarding the $m_i$ as the zero orders and $p_j+1$
as the pole orders of twisted differentials on irreducible components of the corresponding stable curves). In particular, 
every level graph has only finitely many compatible twists. For a graph
of compact type, there exists a unique twist~$\bfp$ if the entries 
of $(\mu[0],\mu[\infty])$ satisfy the condition that  
$\sum_{i=1}^n m_i - \sum_{j=1}^k (p_j+1) = 2g-2$. 
\par
\begin{Defi}\label{def:RT}
Let $1\leq i< j\leq n$. A {\em stable rooted tree} (or simply a {\em rooted tree}) is a twisted
level graph $(\Gamma,\ell,\bfp)$ of compact type satisfying the following conditions: 
\begin{itemize}
\item[i)] One vertex $v_j$ carries the $i$-th and $j$-th legs and no other
of the first~$n$ legs; 
(The vertex $v_j$ is called the {\em root}, a vertex on the path from~$v$ to the root is called an {\em ancestor} of~$v$, and a vertex whose ancestors contain~$v$  is called a~{\em descendant} of $v$.)
\item[ii)] There are no horizontal edges; 
\item[iii)] A vertex $v$ is on level $0$ if and only if $v$ is a leaf.  If $v$ is not a leaf, then 
$\ell(v) = {\rm min}\{\ell(v')\,|\, v'\ \mbox{is a descendant of}\ v\}-1$;
\item[iv)] All vertices of positive genus are leaves and hence on level $0$; 
\item[v)] Each vertex of genus zero other than $v_j$ carries exactly
one of the first $n$ legs.
\end{itemize}
\end{Defi}
Since the root is an ancestor of any other vertex, by definition it is
the unique vertex lying on the bottom level, hence it has genus zero.
Moreover, it is easy to see from the definition that any path towards
the root is strictly going down.  
\par
\begin{figure}[ht]
\begin{tikzpicture}
\tikzstyle{every node}=[font=\scriptsize]
\draw[] (-2,1.72) node[] {$\ell =~0$};
\draw[] (-2,1.1) node[] {$\ell =-1$};
\draw[] (-2,.65) node[] {$\ell =-2$};
\draw[] (-2,.2) node[] {$\ell =-3$};

\path[name path=line A,draw] 
	(-1,1.6) node[shape=circle,draw,inner sep=.5pt,xshift=-1pt,yshift=3.5pt] 
	{$5$} -- +(-55:2) node[xshift=2pt,yshift=-5pt] {$2$}; 
\path[name path=line D,draw]
	(1,1.6) node[shape=circle,draw,inner sep=.5pt,xshift=1pt,yshift=3.5pt] 
	{$1$} -- +(-125:2) node[xshift=-2pt,yshift=-5pt] {$1$}; 
\path[name path=line C,draw] 
	(.4,1.6) node[shape=circle,draw,inner sep=.5pt,xshift=1pt,yshift=3.5pt] 
	{$2$} -- +(-125:1.6) node[xshift=-2pt,yshift=-5pt] {$4$}; 
\path[name path=line B,draw] 
	(-.2,1.6) node[shape=circle,draw,inner sep=.5pt,xshift=1pt,yshift=3.5pt] 
	{$3$} -- +(-125:1.2) node[xshift=-2pt,yshift=-5pt] {$3$};

\fill[name intersections={of=line A and line B}] (intersection-1) circle (1.5pt);
\fill[name intersections={of=line A and line C}] (intersection-1) circle (1.5pt);
\fill[name intersections={of=line A and line D}] (intersection-1) circle (1.5pt);

\draw[] (1.09,1.6) -- +(-50:.15) node[xshift=2pt,yshift=-2pt] {$6$}; 
\draw[] (1.045,1.6) -- +(-90:.15) node[yshift=-4pt] {$7$};
\draw[] (.49,1.6) -- +(-50:.15) node[xshift=2pt,yshift=-2pt] {$5$};
\end{tikzpicture} 
\caption{A rooted tree of genus eleven with three vertices of genus zero (black) 
and seven legs} \label{cap:RT}
\end{figure}
\par
We denote by ${\rm RT}(g,\mu[0],\mu[\infty])_{i,j}$ the set of such 
rooted trees, and sometimes simply by ${\rm RT}(\mu[0],\mu[\infty])_{i,j}$ if $ g = 0$.   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The sum over rooted trees} \label{sec:HurRT}
%%%%%%%%%%%%%%%%%%%%%

Now we assume that $g=0$. Below we define the local contributions from rooted trees 
in Proposition~\ref{prop:intpsiPP1} and complete its proof. Consider a graph $\Gamma \in {\rm RT}(\mu[0],\mu[\infty])_{1,2}$. 
Since by assumption every vertex of $\Gamma$ has genus zero, condition~v) implies that $\Gamma$ has
exactly $n-1$ vertices and $n-2$ edges. Denote by $v_2, \ldots, v_n$ the vertices of $\Gamma$ such that 
 $v_i$ carries the $i$-th leg $h_i$ for $3\leq i\leq n$ and $v_2$ carries the first two legs. This convention is consistent
with our previous notation for the root. We denote by $\mu[\infty]_i$ the list of negative twists at half-edges adjacent to $v_i$. 
These half-edges are either part of the whole edges joining $v_i$ to its descendants (as adjacent vertices to $v_i$ on higher level) or part of the $k$ last legs (corresponding to the $k$ marked poles).
\par
If $i\neq 2$, then there is a unique (non-leg) half-edge $\widetilde{h}_i \neq h_i$
adjacent to~$v_i$ such that $\widetilde{m}_i:=\bfp(\widetilde{h}_i)-1\geq 0$.
Namely, this half-edge is part
of the whole edge joining $v_i$ to its ancestor (as the adjacent vertex to $v_i$ on lower level). 
With this notation we define
the contribution of the rooted tree $\Gamma$ as
\ba\label{eq:h(gamma)}
h(\Gamma) \= h_{\proj^1}((m_1,m_2), \mu[\infty]_2) \,\cdot \,
\prod_{i=3}^{n} h_{\proj^1}((m_i, \widetilde{m}_i),\mu[\infty]_i)\,.
\ea
\par
Let $J\subset [\![1,n+k]\!]$ be a subset such that the cardinalities of $J$ and $J^c$ are at least two. 
Denote by $\delta_J$ the class of the boundary divisor of
$\oM_{0,n+k}$ parameterizing curves that consist of a component with
the markings in $J$ union a component with the markings in $J^{c}$. 
We need the following classical result (see e.g.~\cite[Lemma~7.4]{acgh2}).  
\par
\begin{Lemma} For all $3\leq i\leq n + k$, the following relation of divisor classes holds on $\oM_{0,n+k}$:  
$$
\psi_i \=\sum_{i\in J\subset [\![ 3,n+k]\!]} \delta_J\,.
$$
\end{Lemma}
\par
If $J$ is a subset of $[\![3,n]\!]$, then we denote by $\widetilde{\delta}_{J}=\sum_{J'\subset [\![n+1,n+k]\!]} \delta_{J\cup J'}$.
For $3\leq i \leq n$, the above lemma implies that 
\begin{equation}\label{eqn:psi0}
\psi_i \=\sum_{i\in J\subset [\![ 3,n]\!]} \widetilde{\delta}_J\,.
\end{equation}
\par
We also need the following result about the boundary divisors of $\BH_{\proj^1}(\mu [0],\mu [\infty ])$. 
\par
\begin{Lemma}\label{lem:hurwitz-boundary}
There is a bijection between the boundary divisors of $\BH_{\proj^1}(\mu [0],
\mu [\infty ])$ and the corresponding bi-colored graphs (i.e. with two levels
only). Moreover, a boundary divisor drops dimension under the source
map $f_S$ if its bi-colored graph has more than two vertices.
\end{Lemma}
 \par
 \begin{proof}
 The first part of the claim follows from the same argument as in~\cite[Proposition 7.1]{SauvagetClass}. Here the vertices of level $0$ in the bi-colored graphs correspond to the stable components of the admissible covers that contain the marked poles. For the other part, suppose that a generic point of a boundary divisor has at least two vertices on level $0$ (or on level $-1$). Then one can scale one of the two functions that induce the covers on the two vertices such that the domain marked curve is fixed while the admissible covers vary. It implies that $f_S$ restricted to this boundary divisor has positive dimensional fibers.    
 \end{proof}
\par
\begin{proof}[Proof of Proposition~\ref{prop:intpsiPP1}, case $n \geq 3$]
We will prove the result by induction on $n$. The initial case
$n=2$ follows from the definition of $h(\Gamma)$ in~\eqref{eq:h(gamma)} and
we have also described it explicitly in Section~\ref{subsec:n=2}.    
The strategy of the induction for higher $n$ is by successively
replacing the $\psi_i$ in~\eqref{eq:HNdef} with the sum
over boundary divisors as in the preceding lemma, starting with $i=3$. To
simplify notation, we write $\BH(\mu [0],\mu [\infty ])$ instead of
$\BH_{\proj^1}(\mu [0],\mu [\infty ])$.  We also simply write~$\psi$ and~$\delta$
as classes in the Hurwitz space for their pullbacks via~$f_S$.  
\par 
Consider a boundary divisor $\delta_J$ of $\oM_{0,n+k}$ pulled back to $\BH(\mu [0],\mu [\infty ])$, which is a union of certain boundary divisors of $\BH(\mu [0],\mu [\infty ])$. We would like to compute the intersection number $\delta_J\cdot \prod_{i= 4}^n \psi _i$ on $\BH(\mu [0],\mu [\infty ])$.  By Lemma~\ref{lem:hurwitz-boundary} and the projection formula, the only possible non-zero contribution is from the loci in $\delta_J$ whose 
bicolored graphs have a unique edge $e=(h,h')$ connecting two vertices $v_{0}$ and $v_{-1}$ on level $0$ and level $-1$ respectively, such that the last $k$ markings (i.e. the $k$ marked poles) are contained in $v_0$. In this case we can assume that $\bfp(h)> 0$ (and hence $\bfp(h') < 0$ as $\bfp(h) + \bfp(h') = 0$ by definition). The admissible covers restricted to $v_{0}$ and to $v_{-1}$ belong to Hurwitz spaces of similar type, where at the node
(i.e.\ the edge~$e$) the ramification order of the restricted maps is given by $\bfp(h)-1 = -\bfp(h')-1$. It implies that the locus of such admissible covers can be identified with $\BH(\mu[0]^0,\mu[\infty]^0)\times \BH(\mu[0]^{-1},\mu [\infty]^{-1})$, where $\mu[0]^0$ is the part of $\mu[0]$ contained in $v_0$ union with $\bfp(h)-1$, $\mu[\infty]^0 = \mu[\infty]$, $\mu[0]^{-1}$ is the part of 
$\mu[0]$ contained in $v_{-1}$, and $\mu [\infty]^{-1}$ has a single entry $\bfp(h)$.  
\par
By equation~\eqref{eqn:psi0} we have $\psi_3= \sum_{3\in J\subset [\![3,n+k]\!]} \delta_J$.  
Fix a subset $J\subset [\![ 3,n+k]\!]$ such that $3 \in J$.
If the third marking belongs to $v_{-1}$, we claim that  
$$\BH(\mu [0],\mu [\infty ])\cdot \delta_J\cdot \prod_{i= 4}^n \psi _i \= 0\,.$$
To see this, note that the dimension of $\BH(\mu [0]^{-1},\mu [\infty ]^{-1})$ is equal
to $n(\mu [0]^{-1}) - 2$, as $\mu [\infty ]^{-1}$ has only one entry.  
However there are
$n(\mu [0]^{-1})-1$ markings with label $\geq 4$ on $v_{-1}$. Consequently the intersection with the product of those 
$\psi_i$ vanishes on $\BH(\mu [0]^{-1},\mu [\infty ]^{-1})$. 
\par
Therefore, we only need to consider the case when $v_0$ contains the third marking (hence all markings labeled by $J$), and consequently  
$v_{-1}$ contains the first and second markings (hence all markings labeled by $J^c$). In this case we obtain that  
\begin{eqnarray*}
\BH(\mu [0],\mu [\infty ])\cdot \delta_J\cdot \prod_{i= 4}^n \psi _i
\= h_{\proj^1} (\mu [0]^0,\mu [\infty]^0) \cdot h_{\proj^1} (\mu[0]^{-1},\mu [\infty ]^{-1})\,,
\end{eqnarray*}
where $h_{\proj^1}$ is defined in~\eqref{eq:HNdef}, where in the first factor on the right-hand side the $\psi$-product skips the third marking and the marking from the half-edge of $v_0$, and where in the second factor the $\psi$-product skips the first and second markings. 
\par
Now we use the induction hypothesis to decompose the factors
$h_{\proj_1} (\mu [0]^a,\mu [\infty ]^a)$ for $a=0$ and $a=-1$. It leads to a sum over all possible pairs of rooted trees, where the two rooted trees in each pair generate a new rooted tree. More precisely, one rooted tree in the pair contains the markings of $J^c\cup \{h'\}$ whose root $v_2$ carries the first and second markings, the other rooted tree contains the markings in $J\cup \{h\}$ whose root $v_3$ carries the third marking and $h$, and they generate a new rooted tree by gluing the legs $h$ and $h'$ as a whole edge and by using $v_2$ as the new root.  
\par
Therefore, if $J$ is a subset of $[\![3,n]\!]$ such that $3\in J$, then we obtain that 
\begin{eqnarray*}
  \BH(\mu [0],\mu [\infty ])\cdot \widetilde{\delta}_J\cdot \prod_{i= 4}^n \psi _i \=
  \sum_{\begin{smallmatrix}\Gamma \in {\rm RT}(\mu[0],\mu[\infty])_{1,2}, \\ j\in J \Leftrightarrow \ell(v_3)\leq \ell(v_j) \end{smallmatrix}} h(\Gamma)\,,
\end{eqnarray*}
where the sum is over all rooted trees $\Gamma$ such that the descendants of $v_3$ are exactly the vertices $v_j$ for $j\in J\setminus \{3\}$.   
\par
In summary if we write $\psi_3= \sum_{3\in J\subset [\![3,n]\!]} \widetilde{\delta}_J$, then by the above analysis we thus conclude that 
$\BH(\mu [0],\mu [\infty ])\cdot\psi_3\cdot \prod_{i= 4}^n \psi _i$ is equal to the sum of the contributions $h(\Gamma)$ over all rooted trees $\Gamma$.  
\end{proof}

