der Ast
branch
One of the main offshoots that extend directly from the trunk of a tree or from a thicker branch.
Ein starker Ast ist im Sturm vom Baum abgebrochen.
A strong branch broke off the tree in the storm.

branch
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branch
One of the main offshoots that extend directly from the trunk of a tree or from a thicker branch.
Ein starker Ast ist im Sturm vom Baum abgebrochen.
A strong branch broke off the tree in the storm.

The verb 'sich verästeln' (to branch out) is derived from the noun 'der Ast' (the branch). It means that something divides into many smaller branches or paths, like the branches of a tree.
Das Wegenetz im Wald verästelt sich immer mehr.
The network of paths in the forest branches out more and more.
In its literal meaning, 'Ast' is often combined with verbs that describe an action: 'einen Ast absägen' (to saw off a branch), 'auf einen Ast klettern' (to climb onto a branch), or 'von einem Ast fallen' (to fall from a branch).
Die Kinder kletterten auf einen dicken Ast des Apfelbaums.
The children climbed onto a thick branch of the apple tree.
The figurative meaning of 'Ast' is often found in connection with family structures. Signal words here are 'Stammbaum' (family tree), 'Familie' (family), or 'Abstammung' (descent/ancestry).
Die Forschung zu ihrem Stammbaum zeigte, dass ein Ast der Familie nach Amerika ausgewandert war.
The research into her family tree showed that one branch of the family had emigrated to America.
The expression 'sich einen Ast lachen' (to laugh one's head off) is a fixed idiom. The verb is reflexive ('sich lachen'), and 'einen Ast' is a fixed component in the accusative.
Als er die Geschichte erzählte, hat sich die ganze Gruppe einen Ast gelacht.
When he told the story, the whole group laughed their heads off.
stick / floor
'der Stock' and 'der Ast' both belong to the area of trees and wood, but they are not the same thing. 'der Ast' is a part of a tree that grows out of the trunk or out of a thicker branch. 'der Stock' is usually a long, thin piece of wood that you can use or hold in your hand. So a branch can later become a stick if it has broken off or been cut off. The words are easy to confuse because both can be made of wood. On the tree, you say 'Ast'; as a separate piece of wood or a walking aid, you are more likely to say 'Stock'.

Ein dicker Ast hängt nach dem Sturm gefährlich über der Straße.
After the storm, a thick branch is hanging dangerously over the road.
Im Park hebt Tom einen Stock auf und zeichnet damit Linien in den Sand.
In the park, Tom picks up a stick and uses it to draw lines in the sand.