Extremal values of the trace norm over oriented trees

The trace norm of the digraph D   is defined as N(D)=∑i=1nσi, where σ1≥σ2≥⋯≥σn≥0σ1≥σ2≥⋯≥σn≥0 are the singular values of the adjacency matrix A of D  , i.e. the square roots of the eigenvalues of AA⊤AA⊤. We find the extremal values of NN over the set of oriented trees and over the set of oriented paths.