There are 10/100 * 4000 = <<10/100*4000=400>>400 spruces in the forest
There are 13/100 * 4000 = <<13/100*4000=520>>520 pines in the forest.
There are as many oaks as pines and spruces together, so there are 400 + 520 = <<400+520=920>>920 oaks.
Therefore there are 4000 - 920 - 520 - 400 = <<4000-920-520-400=2160>>2160 birches.
#### 2160