It's a DP on tree problem.

To solve the problem, you need to calculate maximal number of symbols from prefix of string S you have visited on some path started somewhere in subtree of vertex V and ended in V and similarly for suffix (assume they are dp[v][0] and dp[v][1]).

Than, when you calculated this values for all children of some vertex V, you should check is there exists the path passing through the vertex V that is the answer, you need to check is there exists such two children ch1 and ch2 of vertex V, that dp[ch1][0] + f(edge(ch1, v)) + dp[ch2][1] + f(edge(v, ch2)) >= m, where f is 1 or 0, if this edge contains needed symbol of S or not. (Such two children can be found using prefix maximum of dp values)

And if you maintain vertices, where path for dp[v][0] and dp[v][1] starts, all the problem can be done using one simple DFS.