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 Incoming links: Reduced ordered binary decision diagram

A reduced ordered binary decision diagram is obtained by sharing equal terminal nodes, removing a decision node whose two children are equal, and sharing decision nodes with the same tested Boolean variable and the same ordered pair of children. Apply these rules to the binary decision tree in part (c) as follows.

Merge the three zero terminal nodes into one zero terminal node, and merge the five one terminal nodes into one one terminal node.
At $(z, y) = (0, 0)$ the $x$ decision node has children $(0, 0)$ . Remove it and redirect its incoming arrow to zero.
At $(z, y) = (0, 1)$ and $(z, y) = (1, 1)$ the two $x$ decision nodes each have children $(1, 1)$ . Remove both and redirect their incoming arrows to one.
The remaining $x$ decision node, at $(z, y) = (1, 0)$ , has children $(0, 1)$ , so retain it.
The $y$ decision node under $z = 0$ now has children $(0, 1)$ , while the one under $z = 1$ has children $(x, 1)$ . These are different, so neither can be merged or removed. The top $z$ decision node also has different children.

The left panel below is therefore the resulting reduced ordered binary decision diagram. Its full description is

Decision node	Test	Zero child	One child
$r$	$z$	$u$	$v$
$u$	$y$	0	1
$v$	$y$	$w$	1
$w$	$x$	0	1

There are

4 + 2 = 6

The original binary decision diagram also has four decision nodes and two terminal nodes, and is already reduced for

x < y < z

: the two

y

decision nodes have different children and no decision node has two equal children. Thus reversing the order has not made this example smaller. Indeed

y \lor (x \land z)

is unchanged by interchanging

x

and

z

, and the new binary decision diagram is the original one with those two labels exchanged. The uniqueness of a reduced ordered binary decision diagram applies to a fixed variable ordering of a binary decision diagram; it does not assert that every ordering has the same size.