I have two versions for you.
1 The answer you don't like
As <PERSON> suggests, your anemone tree could resemble actual anemones on land. Instead of making the whole treetop one giant set of tentacles with the mouth in the middle, the tree could consist of traditional twigs and branches sporting small tentacle mouths. Each mouth would secrete a sweet smell , be brightly coloured to mimick flowers, or have other means of (passively) attracting prey. This would provide several benefits to the creature/plant:
1.1 Lower Energy consumption
Having the tree move around to catch prey is going to cost energy. A lot of it. Being able to move aroudn superficially while also being rooted underground makes no sense from an evolutionary standpoint, because the increase in range at a rather slow speed does not compensate for the loss of energy. An rooted organism capable of any noticeable controlled movement would either loose this capability, for example if the ground supplied enough nutrients, or loose the roots, in case the movement made up the central aspect of the organism's energy acquisition cycle.
1.2 More Consistent food supply
This is in response to some of the other answers submitted, where the tree preys on larger animals.
Predators that prey on large animals invest a lot of energy in hope of a lot of food (hyenas, wolves, lions), and will spend most of their time conserving energy if there is no food source in sight. Predators that hunt smaller animals (frogs, otters, smaller birds of prey) spend most of their time hunting, since each food ration they acquire will not last as long, but also is not as hard to get.
Since the anemone tree is most likely stationary, it can't search for food; it has to wait for it. As large animals tend to be rarer, I deem it unlikely that such a tree's prey spectrum would consist of anything larger than insects, rodents, birds, small monkeys and the likes. This brings us to the third point:
1.3 Higher likelihood of actually catching anything
As we've ruled out large (and therefore slow) animals, we have to acknowledge that out prey will be fast, faster than out tree can afford to move. 445
Nature shows us that carnivores don't neccessarily have to be fast; there are a number of flesh-eating plants which rely on stickyness, stunning or sedation of the prey upon first contact to keep it in a place where the plant can easily digest it.
2 The answer you might like a little better
So, as the question explicitly asks for moving branches, we have to incorporate them somehow, despite their high energy consumption and low evolutionary likelihood. What benefits could the anemone tree gain from being able to move its branches, when it is a "passive predator"?
2.1 Fishing
While eerily waving vines might deter prey on land, it can be quite the attraction under water. Imagine a tree literally applying a longlining technique with its vines.
2.2 Photosynthesis
There might be not enough prey around, and in case the anemone tree is an actual plant, it could use it's movement capabilities to gain the most sunlight for photosynthesis, the same way people earlier thought sunflowers would do it.
2.3 Very slow prey
Its prey consists solely of slow and/or relatively blind animals like snails, worms, or turtles, so that the tree could graze a larger area for potential food. It's debateable whether a strand of sticky vines could retract with enough force to lift something as large as a turtle. If not, the question arises how the tree could access the food source after killing or immobilizing it with toxins; dragging it into the ground is hardly possible with anything but very soft soil, and would leave visible marks that would warn other prey. Leaving the carcass on the ground would mean leaving it to scavengers, and loosing a big part of its nutrients.
2.4 Symbiotic host
It has a symbiotic relationship with a species that can somehow supply massive amounts of energy for the tree. An example would be ants or bugs decomposing fallen prey. How the tree would benefit from the bugs taking the food is unclear though, as I highly doubt the feasibility of feeding nodules.
2.5 Free movement
Its actually not stationary. The anemone tree is by no means a plant and can easily pull its "roots" from the ground and move freely. Maybe the environment has a high risk of storms or floods partially, so that there would be a reason for the "tree" to lock itself in a stable position. 445