science

Multitree

Multitree is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Multitree rather than just read about it. In short: In combinatorics and order theory, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which there is at most one directed path between any two vertices, or equivalently in which the subgraph reachable from any vertex induces an undirected tree, or a partially ordered set (poset) that does not have four items a, b, c, and d forming a diamond suborder with a ≤ b ≤ d and a ≤…

Key takeaways

Multitree belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Multitree to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Multitree from memory before moving on to harder problems.

Reference excerpt

In combinatorics and order theory, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which there is at most one directed path between any two vertices, or equivalently in which the subgraph reachable from any vertex induces an undirected tree, or a partially ordered set (poset) that does not have four items a, b, c, and d forming a diamond suborder with a ≤ b ≤ d and a ≤ c ≤ d but with b and c incomparable to each other (also called a diamond-free poset). In computational complexity theory, multitrees have also been called strongly unambiguous graphs or mangroves; they can be used to model nondeterministic algorithms in which there is at most one computational path connecting any two states. Multitrees may be used to represent multiple overlapping taxonomies over the same ground set. If a family tree may contain multiple marriages from one family to another, but does not contain marriages between any two blood relatives, then it forms a multitree.

Equivalence between DAG and poset definitions In a directed acyclic graph, if there is at most one directed path between any two vertices, or equivalently if the subgraph reachable from any vertex induces an undirected tree, then its reachability relation is a diamond-free partial order. Conversely, in a diamond-free partial order, the transitive reduction identifies a directed acyclic graph in which the subgraph reachable from any vertex induces an undirected tree.

Diamond-free families A diamond-free family of sets is a family F of sets whose inclusion ordering forms a diamond-free poset. If D(n) denotes the largest possible diamond-free family of subsets of an n-element set, then it is known that

2 ≤ lim n → ∞ D ( n ) / ( n ⌊ n / 2 ⌋ ) ≤ 2 3 11 {\displaystyle 2\leq \lim _{n\to \infty }D(n){\Big /}{\binom {n}{\lfloor n/2\rfloor }}\leq 2{\frac {3}{11}}} , and it is conjectured that the limit is 2.

Related structures A polytree, a directed acyclic graph formed by orienting the edges of an undirected tree, is a special case of a multitree. The subgraph reachable from any vertex in a multitree is an arborescence rooted in the vertex, that is a polytree in which all edges are oriented away from the root. The word "multitree" has also been used to refer to a series–parallel partial order, or to other structures formed by combining multiple trees.

References

Illustrations

Worked examples

Example 1 — a first encounter with Multitree

Start with the simplest possible case. Write down what Multitree claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Multitree before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Multitree ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Multitree

In research: Multitree appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Multitree in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Multitree is common in secondary-school and first-year university syllabi. It links to neighbouring topics Directed acyclic graphs, Order theory, so understanding it makes those chapters shorter.
In everyday life: Look for Multitree outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Multitree in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Multitree means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Multitree out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multitree in simple terms?

Why does Multitree matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Multitree?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Multitree.