The lava and pyroclastic flows are pretty directional, traveling down the path of least resistance. Trees and pretty much anything is going to go up in smoke. And, the rest of the mountain will be not burnt, so trees might still be alive.
I think that depends if the eruption blew the top or side of the mountain off, then you'll have massive amounts of dust and rock raining down all over the area. Those might kill trees but not ignite them since the dust has time to cool as it settles down to earth. Hot ash adds to the force of the uplift by heating the air, so it gets carried up very high. It's only after everything is relatively cooler that it starts to come down -- ash that is.
Heavy material rock, pebbles, dirt will come down immediately and can be hot enough to ignite fires. If the trees are strong and healthy, and there isn't a build up dead wood, limbs, etc, any fire that starts will burn off the ground cover and might not kill the trees. The dust will coat the leaves and the heat can kill them off. But, any trees that have enough energy stored in their root ball will be able to try and regrow leaves next season.
The initial shock wave can cause soil liquefaction which can result in trees momentarily having roots in effectively liquid and they can fall over