Expected number of common edges for a given tree with any other tree
Let $T$ be your reference tree (on $n$ vertices), and let $R$ be the random tree. Label the edges of the tree randomly from $1$ to $n-1$. Let $X_i$ denote the event that the $i$th edge of $R$ is an edge of $T$. Linearity of expectation shows that the expected size of $T \cap R$ is $\sum_i \Pr[X_i]$. Each $X_i$ is a uniformly random edge, and so it belongs to $T$ with probability $|T|/\binom{n}{2} = 2/n$. Hence the expected size of the intersection is $(n-1) \cdot 2/n = 2(1-1/n)$.