The Axe of the Stream is one of Thaumcraft's "elemental tools", offering rapid cutting of trees from ground level, and automatic collection of the wood.
When used on a tree this axe will magically chop the topmost blocks of the tree first, effectively cutting it from the top down. More specifically, it will chop the furthest wood block connected by leaves and wood to the block you are chopping. As well, the wood will automatically "flow" toward the player, and (space permitting) into their inventory. The leaf blocks will also decay more quickly, though saplings do not stream toward the player.
Be careful in dense forests -- if the trees overlap, the axe can end up chopping down a whole section of forest. It can also work on other assemblies of logs, such as the borders of village farms or parts of some houses, and the effect can cross a one-block gap between logs.
The "Streaming" effect harkens back to Thaumcraft 3, but was only restored in version 4.2.