Is this optimum travelling problem under deadlines NP-hard on trees? One of my friends asks me the following scheduling problem on tree. I find it is very clean and interesting. Is there any reference for it?

Problem:
There is a tree $T(V,E)$, each edge has symmetric traveling cost of 1. For each vertex $v_i$, there is a task which needs to be done before its deadline $d_i$. The task is also denoted as $v_i$. Each task has the uniform value 1. The processing time is 0 for each task, i.e., visiting a task before its deadline equals finishing it. Without loss of generality, let $v_0$ denote the root and assuming there is no task located at $v_0$. There is a vehicle at $v_0$ at time 0. Besides, we assume that $d_i \ge dep_i$ for every vertex, $dep_i$ stands for the depth of $v_i$. This is self-evident, the vertex with deadline less than its depth should be taken as outlier. The problem asks to find a scheduling which finishes as many tasks as possible.

Progress: 


If the tree is restricted to a path, then it is in $\mathsf{P}$ via dynamic programming. 
If the tree is generalized to a graph, then it is in $\mathsf{NP}$-complete.
I have a very simple greedy algorithm which is believed 3-factor apporoximation. I have not proved it completely. Rightnow, I am more interested about the NP-hard results. :-)


Thanks for your advice.
For this to be true, we have to make some assumptions. See comments below

For the tree case, I believe that there is a polynomial time recursive dynamic programming algorithm parameterized by the maximum deadline time. The sub-problems are: if we enter a sub-tree by time $t_a$ and exit the sub-tree by time $t_b$, what is the maximum number of tasks we can finish in the sub-tree? The base cases at the leaves are easy and we memoize from the bottom up. 

If we truly parameterized by the maximum deadline time, then the algorithm would not really be polynomial in the size of the tree. However, the length of the longest path that visits every node in the tree is only polynomial in $|V|$, and we never need to check deadline times later than that.