What are the differences between parse and decision trees?
A decision tree is an algorithm for computing a function, in your case the truth value of the formula. In your setting, each internal node of the decision tree is labeled by a variable, and it has two children; you go to the left one if the variable is false, and to the right one if it is true. Leaves are labeled by True or False, and this is the returned value.

For example, one decision tree for $a \land b$ is as follows. The root is labeled $a$. Its left child is a leaf labeled False. Its right child $x$ is labeled $b$. The left child of $x$ is a leaf labeled False, and the right child of $x$ is a leaf labeled True.