In the self-stabilizing algorithmic paradigm for 
distributed computation each node has only a local
view of the system, yet the goal is for the system
to converge to a global state satisfying some
desired property in a finite amount of time.
Typically, when the underlying system is an
arbitrary graph, finding an optimal solution for
the desired property is not possible in a
provably finite amount of time.  This talk
describes two new self-stabilizing algorithms
that optimize their criteria for tree graphs
in a finite (polynomial) amount of time.
The first finds a maximum matching in a tree.
(Previous self-stabilizing algorithms for constructing
maximal, but not necessarily maximum, matchings in a graph
exist.) The second new self-stabilizing algorithm
finds a minimum sized total vertex cover of a tree.