The Euler--Sombor index is a degree-based graph invariant obtained by summing the weights \( \sqrt{d(u)^2+d(v)^2+d(u)d(v)} \) over all edges \( uv \). This paper studies exact degree-transfer formulae for trees and uses them to refine extremal results for fixed matching number, fixed pendent vertices, and fixed diameter. The problem is to identify which local changes force the index to increase, and how the extremal values vary when the parameters change. The method replaces global grafting arguments by verifiable identities for \( \omega(x,y)=\sqrt{x^2+y^2+xy} \). We prove increment formulae, subdivision and leaf-transfer identities, and a strict branch-concentration theorem. These tools give closed expressions, equality cases, monotonicity laws, structure analysis, and stability gaps for starlike trees, brooms, matching trees, and one-hub diameter trees. The numerical tables correct representative computations and make each edge-degree count reproducible. Several open problems for chemical trees are stated.