# Node Types

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Node Types [1]

Node Types are classifications of nodes as expected inside a minimal alpha-beta tree, or as determined by the search. These types were first formulated by Donald Knuth and Ronald Moore when describing the alpha-beta algorithm [2] and further analyzed by Judea Pearl [3]. While Knuth used the terms Type 1, 2, and 3, Tony Marsland and Fred Popowich introduced the more descriptive terms PV-, Cut- and All-nodes [4] [5].

Node types as established by the search are stored inside the transposition table, indicating whether the score is either exact, lower or upper bounded. Expected node types, as determined by tree topology, probing the transposition table, or comparing scores of a static evaluation considering threats, or even a reduced search or quiescence search, with the bounds, may be considered by various (parallel) search algorithms and in decisions concerning selectivity.

# PV-Nodes

PV-nodes (Knuth's Type 1) are nodes that have a score that ends up being inside the window. So if the bounds passed are [a,b], with the score returned s, a<s<b. These nodes have all moves searched, and the value returned is exact (i.e., not a bound), which propagates up to the root along with a principal variation.

# Cut-Nodes

Cut-nodes (Knuth's Type 2), otherwise known as fail-high nodes, are nodes in which a beta-cutoff was performed. So with bounds [a,b], s>=b. A minimum of one move at a Cut-node needs to be searched. The score returned is a lower bound (might be greater) on the exact score of the node

• In Principal Variation Search Cut-nodes are searched with null windows
• The child of a Cut-node is an All-node
• The parent of a Cut-node is either a PV- or All-node
• The ply distance of a Cut-node to its PV-ancestor is odd

# All-Nodes

All-nodes (Knuth's Type 3), otherwise known as fail-low nodes, are nodes in which no move's score exceeded alpha. With bounds [a,b], s<=a. Every move at an All-node is searched, and the score returned is an upper bound, the exact score might be less.

• In Principal Variation Search All-nodes are searched with null windows
• The children of an All-node are Cut-nodes
• The parent of an All-node is a Cut-node
• The ply distance of an All-node to its PV-ancestor is even

# Node Types in PVS

## Onno

Onno by Onno Garms determines expected node types, beside other things, to perform IID not only at PV-nodes but also at expected Cut-nodes. Onno Garms gave the following rules [6]

• The root node is a PV-node
• The first child of a PV-node is a PV-node
• The further children are searched by a scout search as CUT-nodes
• PVS re-search is done as PV-node
• The first node of bad pruning is a CUT-node
• The node after a null move is a CUT-node
• The first node of null move verification is a CUT-node
• Internal iterative deepening does not change the node type
• The first child of a CUT-node is an ALL-node
• Further children of a CUT-node are CUT-nodes
• Children of ALL-nodes are CUT-nodes

## Summary

Following summary was given by Pradu Kannan similar to Onno's definition as an attempt to predict the node type before searching the node by assuming reasonably good move ordering on PV-nodes and Cut-nodes [7]:

• The root node is a PV-node
• The first child of a PV-node is a PV-node
• Children of PV-nodes that are searched with a zero-window scout search are Cut-nodes
• Children of PV-nodes that have to be re-searched because the scout search failed high, are PV-nodes
• The first child of a Cut-node, and other candidate cutoff moves (nullmove, killers, captures, checks, ...) is an All-node
• A Cut-node becomes an All-node once the first and all candidate cutoff moves are searched
• Children of All-nodes are Cut-nodes

# Number of Leaf Nodes

## Formulas

As emphasized by Daniel Shawul [8], the formulas for the number of leaf nodes of a certain Node type with uniform depth n and branching factor b, using floor and ceiling of n/2 (cut the ceiling, all the floor) ...

```PV  = 1
CUT = b⌈n/2⌉ - 1
ALL = b⌊n/2⌋ - 1
```

... are already the subexpressions ...

```L = PV + CUT + ALL
```

... in Levin's famous formula demonstrating the odd-even effect of alpha-beta

```L = b⌈n/2⌉ + b⌊n/2⌋ - 1
```

## Table

Assuming a constant branching factor of 40, this results in following number of leaves:

number of leaves with depth n and b = 40
depth worst case best case PV CUT ALL
n bn b⌈n/2⌉ + b⌊n/2⌋ - 1 1 b⌈n/2⌉ - 1 b⌊n/2⌋ - 1
0 1 1 1 0 0
1 40 40 1 39 0
2 1,600 79 1 39 39
3 64,000 1,639 1 1,599 39
4 2,560,000 3,199 1 1,599 1,599
5 102,400,000 65,569 1 63,999 1,599
6 4,096,000,000 127,999 1 63,999 63,999
7 163,840,000,000 2,623,999 1 2,559,999 63,999
8 6,553,600,000,000 5,119,999 1 2,559,999 2,559,999

# Quotes

Robert Hyatt on the distinction between ALL- and CUT-nodes [9]:

```In CB, I used this extensively, because you never want to split at a CUT node, and want to prefer an ALL node. YBW tries to recognize an ALL node by requiring that at least one move be searched before splitting, since > 90% of CUT nodes are discovered on the first move searched. But waiting on that first move introduces some wait time, and if you knew it was an ALL node from the get-go, you could split before the first move is completed. That was the basic premise behind the iterated search and DTS algorithm in Cray Blitz ...
```

# Forum Posts

## 2000 ...

Re: Fruit - Question for Fabien by Fabien Letouzey, CCC, March 11, 2004