What are limitations of decision tree approaches to data analysis?
Simple decision trees have some limitations listed below. Fortunately, some of these can be fixed used ensemble learning techniques (think bagging, boosting...).
Concerning limitations :

Trees tend to overfit quickly at the bottom. If you have few observations in last nodes, poor decision can be taken. In this situation, consider reducing the number of levels of your tree or using pruning.

Trees can be unstable because small variations in the data might result in a completely different tree being generated.

Decision trees perform greedy search of best splits at each node. This is particularly true for CART based implementation which tests all possible splits. For a continuous variable, this represents 2^(n-1) - 1 possible splits with n the number of observations in current node.

For classification, if some classes dominate, it can create biased trees. It is therefore recommended to balance the dataset prior to fitting.
Also, Some distributions can be hard to learn for a decision tree. An example below (XOR) :