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Expressions as Trees

Here is an expression in full form.
In[1]:=
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Out[1]//FullForm=
TreeForm prints out expressions to show their "tree" structure.
In[2]:=
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Out[2]//TreeForm=
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.
In[3]:=
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Out[3]//TreeForm=
Here is a more complicated expression.
In[4]:=
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Out[4]=
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.
In[5]:=
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Out[5]//TreeForm=
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.