Tree vs lightning rod: why does one burn and the other not?

The amount of heat generated by current flowing through a resistor (whether from lightning or more ordinary sources) is directly related to the power dissipated by the resistor, which is $$ P = I^2 R.$$ $R$ is small for objects made from good conductors, which many metals are, and large for objects that are made from bad conductors like plastic or wood. Since a lightning strike has a very short duration, the total heat generated during such a strike is not enough to melt metal, but enough to set wood aflame or melt plastic.

If you let the large currents from the lightning strike run through the metal for longer, it probably would also heat up gradually and eventually melt, but this would take longer than the time scale on which lightning occurs.

The high electrical current in a lightning strike delivers heat energy along the full length of the lightning bolt. Part of that length is in the ionized air over the plane, part is the plane's fuselage, and part is the ionized air from the plane down to ground. The current is the same, but the heat generated is proportional to the electrical resistance of the path, and the aluminum of the plane (as well as the joints that hold the aluminum parts together) has very low electrical resistance. So, the air path gets very hot, while the aluminum path does not.

Trees and air have high electrical resistance, but (at breakdown) have a narrow conducting channel (ionized gas is 'Z-pinched' into a narrow channel, and the first woody parts that carbonize will hog all the current). So,

$$HeatPower = I^2 R = I^2 \rho * L/A$$

Heating per unit length of air is more than the aluminum plane because air has higher resistivity, even with comparable conduction cross-sectional area.