Square Root of LCA Convolution

You are given a rooted tree with $N$ vertices (numbered $1$ through $N$) and a prime number $P$. Vertex $1$ is the root of the tree and for each $v$ ($2 \le v \le N$), the parent of vertex $v$ is $p_v$ ($p_v \lt v$). A vertex $s$ is an ancestor of another vertex $f$ if $s$ lies on the path between $1$ and $f$ (in particular, $f$ is its own ancestor). For two vertices $u$ and $v$, let's define $