I think that love takes on a couple of different forms in the poem. There is the obvious love for the tree. It has become the symbol that has endured throughout time, even when the people associated with it has not. There is the love for the playmates that are so closely a part of the memory of the speaker, presumably the poet, but also for the tree and the shared experiences of all of them: Speaker, companions, tree, and being in the world are all linked together. There is a love for the memories that the tree holds that are unable to be produced in the modern setting.
It is here where I think that Dutt's vision of love is one of in the pain of nostalgia. The love of the past, especially one that is past us, is something that the speaker alludes to as something that causes a sense of pain and sweet longing. It is for this reason why the tree holds so much in way of meaning and significance for it is a link to something that is gone and not to be replicated again. The tree is a "carousel" that enables the speaker to visit these memories again and again. In articulating this love for the past and for what once was, the speaker holds love towards the tree and all that it embodies. Consider a vision of nostalgia that is given courtest of Don Draper from the television series, "Mad Men:"
Nostalgia - its delicate, but potent. Teddy told me that in Greek nostalgia literally means 'the pain from an old wound.' It’s a twinge in your heart far more powerful than memory alone. This device isn’t a spaceship, it’s a time machine. It goes backwards, and forwards… it takes us to a place where we ache to go again. It’s not called the wheel, it’s called the carousel. It let’s us travel the way a child travels - around and around, and back home again, to a place where we know are loved.
I think that this quote really captures well the love that Dutt finds in the tree. It is a reminder of the love she experienced with others and the feeling of knowing that she can love and is able to be loved. The tree is this "carousel," and it is for this reason that the love experienced in the poem is "delicate, but poetent" and reflects "the pain from an old wound," which is the past of what can never be in the future.
The poet Toru Dutt had immense love for her siblings who died before her. She spent her childhood with her siblings playing under the Casuarina tree. So whenever the memoir of the poet's childhood and her siblings came before her eyes the majestic image of the tree even came into her remembrance.
Hence, the poet's love for the tree is because her dead siblings and because of those dulcet memory of her childhood.