  We present a general framework to generate trees every vertex of which has a
non-negative weight and a color. The colors are used to impose certain
restrictions on the weight and colors of other vertices. We first extend the
enumeration algorithms of unweighted trees given in [19, 20] to generate
weighted trees that allow zero weight. We avoid isomorphisms by generalizing
the concept of centroids to weighted trees and then using the so-called
centroid-rooted canonical weighted trees. We provide a time complexity analysis
of unranking algorithms and also show that the output delay complexity of
enumeration is linear. The framework can be used to generate graph classes
taking advantage of their tree-based decompositions/representations. We
demonstrate our framework by generating weighted block trees which are in
one-to-one correspondence with connected block graphs. All connected block
graphs up to 19 vertices are publicly available at [1].
