Embedding a n-tree into a b-dimensional space
This is not possible if the dimension $m$ is fixed. Consider a complete $b$-ary tree, in which all edges are oriented from the root $r$. Then all vertices lie in a ball of radius $h$ around $v_r$. On the other hand, the distance between every two leaves is greater than $h$. Since there are $N = b^h$ leaves, we get that 


  there are $N$ points in a ball of radius $h$ such that all the pairwise distances between the points are greater than $h$


We must have that $m = \Omega(\log N) = \Omega(h \log b)$.