Expressions as Trees
Here is an expression in full form.
prints out expressions to show their "tree" structure.
You can think of any Mathematica expression as a tree. In the expression above, the top node in the tree consists of a Plus. From this node come two "branches", and . From the node, there are then two branches, and , which can be viewed as "leaves" of the tree.
This matrix is a simple tree with just two levels.
Here is a more complicated expression.
The tree for this expression has several levels.
The indices that label each part of an expression have a simple interpretation in terms of trees. Descending from the top node of the tree, each index specifies which branch to take in order to reach the part you want.