Yeah I wouldn't say that trees are not affected by it but that they are 'generally' more robust than most other methods like a linear model. This is due to how they handle features by assessing each and accessing only one at a time to generate a split (assuming we are talking about CART). So you could have 100 features for 50 observations but a tree may only use 3 of those features to generate a prediction, they do not 'ignore' the other features but they do not use them for fitting.
This is in stark contrast to a linear regression which would generate a coefficient for each feature or a neural net that would generate a ton of parameters for each.
It's that overparameterization that can get us into trouble and trees are just better at handling it, although there are specific implementations of NNs and LRs which are better than the standard methods.
So, in short (this is a generalization of course):
If you are in the situation where you have way more features than observations then all methods benefit from doing analysis to eliminate less-than-useful features, trees just don't benefit as much as most other methods...generally