Must the leaves be bound together:[1]   

It is a Mitzvah Min Hamuvchar to purchase a Lulav which its leaves have not separated at all from the spine[2] and are thus completely bound.[3] If the leaves began hardening and separating from the spine, it is valid so long as the leaves are still potentially able to be bound to the spine.[4] If majority[5] of the leaves have separated to the point that they have hardened and can no longer be joined to the spine of the Lulav[6], it is invalid.[7]

Leaves that bend downwards:[8] If the majority[9] of the leaves of the Lulav do not rise together with the spine of the Lulav and rather bend downwards below the spine, then the Lulav is Biblically invalid.[10] This applies even if one binds the leaves onto the spine using string and the like.[11] This applies even if the leaves have not completely separated from the spine.

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[1] 645/3-7

[2] This means that they have not separated from their growth on the spine to the point that they are not resting on the spine, even if the leaves are still growing upwards. [ibid]

[3] 645/5

[4] 645/4

The reason: The reason for why the leaves do not have to be actually bound to the spine so long as they are potentially bound is because the Torah does not state Kafus but rather Kapos and hence it is only potentially required to be able to be Kafus. [ibid]

[5] 645/7; Only if majority of the leaves have separated from the spine in a way that they can no longer be bound to the spine is the Lulav invalid. If however only minority of the leaves have split and they can thus still be bound to the spine, and these leaves cover majority of the spine, the Lulav is valid. [ibid]

[6] The leaves of the palm branch begin to harden as they remain on the tree and separate from the spine.[ibid]

[7] The reason: This is learned from the word “Kapos” [branches] which can also be read “Kafus” which means bound.

[8] 645/6

[9] 645/7

[10] The reason: The reason for this invalidation is because the Lulav is not considered Hadar if the leaves droop downwards and it is hence Biblically invalid. [ibid]

[11] The reason for this is because the Lulav is no longer Hadar, and is hence Biblically invalid. [ibid]