[Math] Understanding Abstract Syntax Tree

Question

Benjamin C. Pierce's Types and Programming Languages presents on page 54:

The following expression, (1 + 2) * 3, becomes the following concrete Abstract Syntax Tree:

Pierce goes onto say:

To saving writing too many parentheses, we adopt two conventions when writing lambda-terms in linear form. First, application associates to the left – that is, s t u stands for the same tree as (s t) u:

Second, the bodies of abstraction are taken to extend as far to the right as possible, so that, for example,

$$\lambda x. \lambda y. x y x$$

stands for the same tree as:

$$\lambda x. (\lambda y. ((x y) x)$$

My incomplete understanding is that the second tree can be represented in a Lambda Calculus via the last tree?

Or, put differently, how does the second tree map/relate to the third? I'm not understanding the $\lambda$ usages.

Answer 1

The terms $\lambda x.\lambda y.x y x $ and $\lambda x.(\lambda y.((x y) x))$ both have the same abstract syntax tree, namely

         lambda x
            |
         lambda y
            |
          apply
          /   \
       apply   x
       /   \
      x     y

This means that, as far as the meaning is concerned, the two symbol strings "$\lambda x.\lambda y.x y x $" and "$\lambda x.(\lambda y.((x y) x))$" should be thought of as two representations of the same term.