A compact, connected, locally connected metric continuum that contains no simple closed curve; equivalently, a continuum in which every two points are joined by a unique arc and which exhibits branching points and endpoints.

Principle

A dendrite is organized by the absence of cycles together with local connectedness: forbidding any simple closed curve forces unique arcwise connectivity and a tree-like branching topology within a compact continuum.

Demonstration

Any finite tree considered as a compact 1-dimensional continuum (for example, a finite planar tree embedded in R^2) is a dendrite; more subtle examples include certain Julia sets of complex polynomials that are locally connected and contain no simple closed curve.

Misapplication

Calling a general acyclic graph a dendrite when it is not compact or not locally connected (for instance an infinite graph with accumulation points) misapplies the notion by ignoring compactness and local connectivity requirements.

Consequence

When a continuum is a dendrite it has unique arcs between points, a well-defined set of branch points and endpoints, and no nontrivial loops; many classification and decomposition results (e.g., by order of points) follow from these properties.

Reversal

The reversal is a continuum that contains a simple closed curve (a loop), such as a circle or any continuum with a nontrivial cycle, which violates the defining acyclicity of a dendrite.

Boundary

The term is restricted to compact, connected, locally connected metric continua; noncompact trees, graphs with cycles, and continua that fail local connectivity lie outside the definition.

Semantic Tension

Dendrite versus abstract tree: a tree in graph theory may be finite or combinatorial and need not be a compact continuum, whereas a dendrite is a topological continuum with metric and local connectivity hypotheses—both evoke branching but differ in compactness and topological nuance.

Synthesis

A dendrite is the compact, locally connected, cycle-free continuum whose local and global structure is governed by unique arcs and discrete branching: a topological continuum analogue of a tree with the added continuity and compactness constraints.