This code with unitialised 'root' got crash on 10 test:

    int root;// = 0;
    for (i = 0; i < g.size(); ++i)
        if (g[i].size() == 1)
        {
            root = i;
            break;
        }
where g[i] - vector of adjacent vertices for i-th vertex.
So, graph in 10 test isn't tree.

Sorry :) I forgot about unique vertex.