  The Burning Number Conjecture, that a graph on $n$ vertices can be burned in
at most $\lceil \sqrt{n} \ \rceil$ rounds, has been of central interest for the
past several years. Much of the literature toward its resolution focuses on two
directions: tightening a general upper bound for the burning number, and
proving the conjecture for specific graph classes. In the latter, most of the
developments work within a specific graph class and exploit the intricacies
particular to it. In this article, we broaden this approach by developing
systematic machinery that can be used as test beds for asserting that graph
classes satisfy the conjecture. We show how to use these to resolve the
conjecture for several classes of graphs including triangle-free graphs with
degree lower bounds, graphs with certain linear lower bounds on
$r$-neighborhood sizes, all trees whose non-leaf vertices have degree at least
$4$, trees whose non-leaf vertices have degree at least $3$ (on at least $81$
vertices), trees whose non-leaf vertices are less than $\frac{2}{3}$
concentrated in degree $2$, and trees with a low concentration of high degree
non-leaf vertices (the last two results holding for sufficiently many non-leaf
vertices).

Combinatorics