  A tree diagram is a tree with positive integral weight on each edge, which is
a notion generalized from the Dynkin diagrams of finite-dimensional simple Lie
algebras. We introduce two nilpotent Lie algebras and their extended solvable
Lie algebras associated with each tree diagram. The solvable tree diagram Lie
algebras turn out to be complete Lie algebras of maximal rank analogous to the
Borel subalgebras of finite-dimensional simple Lie algebras. Their abelian
ideals are completely determined. Using a high-order Campbell-Hausdorff formula
and certain abelian ideals of the tree diagram Lie algebras, we solve the
initial value problem of first-order evolution partial differential equations
associated with nilpotent tree diagram Lie algebras and high-order evolution
partial differential equations, including heat conduction type equations
related to generalized Tricomi operators associated with trees.
