Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already declare readOnly/idempotent/non-destructive, and the description adds genuinely new operational facts: the input cap at 30, the output representation switching to a decimal string past 15 digits, and the per-call price. Cost and return-format disclosure are exactly the kind of context annotations cannot carry.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

Four short sentences, front-loaded with the formula and examples before the output-format caveat and the access/pricing note. No sentence is redundant with the schema or annotations.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

For a one-parameter deterministic math tool with no output schema, the description covers domain, edge cases, output representation and cost — nearly everything needed to call it correctly. The only gap is not relating it to the near-identical sibling cayley_rooted.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema coverage is 0%, so the description carries the burden, and it defines n as the number of labeled vertices, bounds it to 1..30, and pins the edge cases (n=1 -> 1, n=2 -> 1). It stops short of stating the type or that n must be an integer, which the schema alone supplies.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

States the exact object computed with a formula (number of labeled trees on n vertices = n^(n-2)) plus worked examples, so the tool's purpose is unambiguous. It does not name the close sibling cayley_rooted, so an agent must infer the difference from the exponent rather than reading it explicitly.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

Usage is implied by the formula and the stated domain 'n runs from 1 to 30', which tells the agent when the tool is valid. However, there is no explicit when-to-use/when-not guidance and no named alternative (e.g. cayley_rooted for rooted trees), leaving selection to inference.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.