A high-strength character (D&D 3.5) attempts to snap a tree or thick branch in half across their knee, or kick a tree down in the forest. This is not dealing damage, this is clearly a Strength check versus some DC, similar to kicking down a door. What is the RAW DC to break a tree? How does this change based on the tree's diameter?
Break DC 25 to 35 depending on diameter
A "Small Tree" appears in Anauroch: The Empire of Shade, p.119. It is one foot thick, with an AC of 4, hardness 5, 150 hit points, and a break DC of 25.
"Dead Trees" appear in Dragon magazine #360, p.55, with the same statistics.
A "Large Tree" appears in Fortress of the Yuan-Ti, page 29. It is five feet thick, with an AC of 3, hardness 5, 600 hit points, and a break DC of 35.
Naturally, the break DC could be higher or lower for trees which are thicker or thinner than normal, respectively.
The Rules Compendium makes the following comment:
BREAKING OBJECTS When a creature tries to break something with sudden force rather than by dealing damage, use a Strength check (including a modifier due to size) to see whether it succeeds. The DC depends more on the construction of the item than on the material.
In other words, it seems that one is finding the weak point in the construction of the composite object, not necessarily actually breaking a given material. Doors are typically built from multiple parts, after all. And doors are the majority of examples in the list.
It then proceeds to give a list of examples, none of which are a tree. However, if you could find something which is close to the construction of a tree on the list, the DM might be willing to adhoc the DC.
The highest DC given is 28, for a stone door, iron door, and masterwork manacles.