  For a tree with the given sequence of vertex degrees the spectral radius of
its terminal distance matrix is shown to be bounded from below by the the
average row sum of the terminal distance matrix of the, so called, BFS-tree
(also known as a greedy tree). This lower bound is typically not tight;
nevertheless, since spectral radius of the terminal distance matrix of BFS-tree
is a natural upper bound, the numeric simulation shows that relative gap
between the upper and the lower bound does not exceed 3% (we also make a step
towards justifying this fact analytically.) Therefore, the conjecture that
BFS-tree has the minimum terminal distance spectral radius among all trees with
the given degree sequence is valid with accuracy at least 97%. The same
technique can be applied to the distance spectral radius of trees, which is a
more popular topological index.
