  In the Dynamic Programming approach to optimal control problems a crucial
role is played by the value function that is characterized as the unique
viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well
known that this approach suffers of the "curse of dimensionality" and this
limitation has reduced its practical in real world applications. Here we
analyze a dynamic programming algorithm based on a tree structure. The tree is
built by the time discrete dynamics avoiding in this way the use of a fixed
space grid which is the bottleneck for high-dimensional problems, this also
drops the projection on the grid in the approximation of the value function. We
present some error estimates for a first order approximation based on the
tree-structure algorithm. Moreover, we analyze a pruning technique for the tree
to reduce the complexity and minimize the computational effort. Finally, we
present some numerical tests.
