Pattern avoidance in trees Lara Pudwell Pattern avoidance in trees Introduction Brief history Contiguous tree patterns Lara Pudwell (Valparaiso University) Definition & examples faculty.valpo.edu/lpudwell Enumeration Non- contiguous patterns Definition & Notre Dame Discrete Math Seminar examples Generating functions November 27, 2012 Sets of tree patterns Connections Partially supported by NSF grant DMS-0851721 to other objects OEIS hits Pattern-avoiding permutations Summary
Outline Pattern Introduction 1 avoidance in trees Brief history Lara Pudwell Contiguous tree patterns 2 Introduction Definition & examples Brief history Contiguous Enumeration tree patterns Definition & examples Non-contiguous patterns 3 Enumeration Definition & examples Non- contiguous Generating functions patterns Definition & Sets of tree patterns examples Generating functions Connections to other objects 4 Sets of tree patterns OEIS hits Connections to other Pattern-avoiding permutations objects OEIS hits Summary Pattern-avoiding 5 permutations Summary
Outline Pattern Introduction 1 avoidance in trees Brief history Lara Pudwell Contiguous tree patterns 2 Introduction Definition & examples Brief history Contiguous Enumeration tree patterns Definition & examples Non-contiguous patterns 3 Enumeration Definition & examples Non- contiguous Generating functions patterns Definition & Sets of tree patterns examples Generating functions Connections to other objects 4 Sets of tree patterns OEIS hits Connections to other Pattern-avoiding permutations objects OEIS hits Summary Pattern-avoiding 5 permutations Summary
History of Tree Patterns: Labelled Trees Pattern 1983: Flajolet and Steyaert avoidance in trees focus on asymptotic probability of avoiding a given pattern Lara Pudwell 1990: Flajolet, Sipala, and Steyaert Introduction every leaf of pattern must be matched by a leaf of the tree Brief history motivated by compactly storing expressions in computer Contiguous tree patterns memory Definition & e.g. d examples sin( x cos 2 ( e x +1 )) � � = Enumeration dx Non- contiguous patterns Definition & examples Generating functions Sets of tree patterns Connections to other objects OEIS hits Pattern-avoiding permutations Summary
History of Tree Patterns: Labelled Trees Pattern 1983: Flajolet and Steyaert avoidance in trees focus on asymptotic probability of avoiding a given pattern Lara Pudwell 1990: Flajolet, Sipala, and Steyaert Introduction every leaf of pattern must be matched by a leaf of the tree Brief history motivated by compactly storing expressions in computer Contiguous tree patterns memory Definition & examples 2012: Dotsenko Enumeration pattern may occur anywhere in tree Non- contiguous motivated by operad theory patterns Definition & examples Generating functions Sets of tree patterns Connections to other objects OEIS hits Pattern-avoiding permutations Summary
History of Tree Patterns: Unlabelled Trees Pattern 2009: Rowland avoidance in trees contiguous pattern avoidance in binary trees Lara Pudwell patterns can be anywhere, not just at leaves Introduction 2010: Gabriel, Peske, P., Tay Brief history extended Rowland’s results to m -ary trees Contiguous tree patterns 2011: Dairyko, P., Tyner, Wynn Definition & examples non-contiguous pattern avoidance in binary trees Enumeration Non- 2012: P., Serrato, Scholten, Schrock contiguous patterns non-contiguous pattern containment in binary/ m -ary trees Definition & examples Generating functions Sets of tree patterns Connections to other objects OEIS hits Pattern-avoiding permutations Summary
Key Question Pattern avoidance in trees Lara Pudwell Today, our trees will be: Introduction Brief history rooted (root vertex at top) Contiguous tree patterns ordered (left child and right Definition & examples child are distinct) Enumeration Non- full binary (each vertex has contiguous patterns exactly 0 or 2 children) Definition & examples Generating functions Sets of tree patterns Connections to other objects OEIS hits Pattern-avoiding permutations Summary
Key Question Pattern avoidance in trees Lara Pudwell Today, our trees will be: Introduction Brief history rooted (root vertex at top) Contiguous tree patterns ordered (left child and right Definition & examples child are distinct) Enumeration Non- full binary (each vertex has contiguous patterns exactly 0 or 2 children) Definition & examples Generating functions Sets of tree patterns Connections to other objects Question: How many trees with n leaves avoid a given tree OEIS hits pattern? Pattern-avoiding permutations Summary
Outline Pattern Introduction 1 avoidance in trees Brief history Lara Pudwell Contiguous tree patterns 2 Introduction Definition & examples Brief history Contiguous Enumeration tree patterns Definition & examples Non-contiguous patterns 3 Enumeration Definition & examples Non- contiguous Generating functions patterns Definition & Sets of tree patterns examples Generating functions Connections to other objects 4 Sets of tree patterns OEIS hits Connections to other Pattern-avoiding permutations objects OEIS hits Summary Pattern-avoiding 5 permutations Summary
Tree patterns Pattern avoidance in trees Lara Pudwell Contiguous tree pattern Introduction Tree T contains tree t if and only if T contains t as a Brief history contiguous rooted ordered subtree. Contiguous tree patterns Definition & Example: examples Enumeration Non- contiguous patterns Definition & examples contains and but avoids . Generating functions Sets of tree patterns Connections to other objects OEIS hits Pattern-avoiding permutations Summary
What is the number a ( n ) of n -leaf binary trees avoiding t ? Pattern avoidance in t = trees Lara Pudwell a ( n ) = 0 Introduction Brief history Contiguous tree patterns Definition & t = examples Enumeration � 1 Non- n = 1 contiguous a ( n ) = patterns 0 n > 1 Definition & examples Generating functions Sets of tree patterns Connections to other t = or t = objects OEIS hits Pattern-avoiding a ( n ) = 1 permutations Summary
What is the number a ( n ) of n -leaf binary trees avoiding t ? Pattern avoidance in trees Lara Pudwell t = Introduction Brief history Contiguous tree patterns Definition & examples Enumeration “Typical” tree avoiding t : Non- contiguous patterns Definition & examples Generating � 1 functions n = 1 Sets of tree patterns a ( n ) = 2 n − 2 n > 1 Connections to other objects OEIS hits Pattern-avoiding permutations Summary
What is the number a ( n ) of n -leaf binary trees avoiding t ? Pattern avoidance in trees Lara Pudwell t = Introduction Brief history Contiguous tree patterns Definition & examples Enumeration Non- “Typical” tree avoiding t : contiguous patterns Definition & examples Generating � 1 functions Sets of tree n = 1 patterns a ( n ) = 2 n − 2 n > 1 Connections to other objects OEIS hits Pattern-avoiding permutations Summary
What is the number a ( n ) of n -leaf binary trees avoiding t ? Pattern avoidance in trees Lara Pudwell t = Introduction Brief history “Typical” trees avoiding t : Contiguous tree patterns Definition & examples Enumeration Non- contiguous patterns Definition & examples Generating Donaghey and Shapiro showed that functions Sets of tree patterns a ( n ) = M n − 1 (Motzkin numbers). Connections to other objects OEIS hits Pattern-avoiding permutations Summary
Contiguous pattern enumeration data Pattern avoidance in a ( n ) t trees Lara Pudwell 0 Introduction � 1 n = 1 Brief history Contiguous 0 n > 1 tree patterns Definition & examples Enumeration 1 Non- contiguous patterns Definition & examples 2 n − 2 Generating functions Sets of tree patterns Connections to other M n − 1 (Motzkin numbers) objects OEIS hits Pattern-avoiding permutations Summary
Contiguous tree pattern enumeration Pattern avoidance in trees Lara Pudwell Rowland Introduction Devised algorithm to find functional equation for Brief history avoidance generating function for any set of tree patterns. Contiguous tree patterns Generating functions are always algebraic. Definition & examples Enumerated trees containing specified number of copies of Enumeration Non- a given tree pattern. contiguous patterns Completely determined equivalence classes for tree Definition & examples patterns with at most 8 leaves. Generating functions For n = 1 , 2 , 3 , . . . , there are 1 , 1 , 1 , 2 , 3 , 7 , 15 , 44 , . . . Sets of tree patterns equivalence classes of n -leaf binary trees. Connections to other objects OEIS hits Pattern-avoiding permutations Summary
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