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Expressions as Trees

Here is an expression in full form.
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TreeForm prints out expressions to show their "tree" structure.
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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", x^3 and (1+x)^2. From the x^3 node, there are then two branches, x and 3, which can be viewed as "leaves" of the tree.
This matrix is a simple tree with just two levels.
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Here is a more complicated expression.
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The tree for this expression has several levels. The representation of the tree here was too long to fit on a single line, so it had to be broken onto two lines.
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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.