The line of the White Trees
First there was the two trees in Valinor Telperion ( the silver tree ) and Laurelin ( the golden tree ), of these Yavanna made an image ( a living tree but not giving lights of it own ) for the Eldar in Tirion and that was named Galathilion. Galathilion was the first of the White Trees and from that came the one at Tol Eressëa named Celeborn. Of Celeborn came Nimloth the Fair of Númenor and from Nimloth the White Tree of Minas Ithil and from that the one in Minas Tirith. The last we know of the White Trees is the young tree Aragorn II found high above Minas Tirith upon mount Mindolluin. None of those last three White Trees has a name we know.
But there was other images of Telperion and Laurelin; Turgon made images of both in Gondolin. Of Telperion he made an image called Belthil and of Laurelin he made one called Glingal. In the Earliest Tales the two images was named Bansil and Glingol and they were not images but were shoots from the Trees of Valinor, they were planted in the Square of the King at each side of the palace doors. In the final version Belthil and Glingal were made in silver and gold and thus not living trees.
Galathilion.
The story of Galathilion, rather the name than the tree itself, is quite confusing. The name was object of several changes towards it`s final application during the evolution of the tales. In the Earliest Tales In the Annals of Aman Galathilion became the name of the image of Telperion Yavanna gave to the Noldor ( but not to the Vanyar it seems since they in the earlier conceptions already [7 Valian years. The years later changed to 25 Valian years] had left Tuna and Tirion, moving from the Calacirya into the main land of Aman ? )
Galathilion was, according to the Quenta Silmarillion, the Noldorin name of Telperion and not of an image of it. There was no such image given by Yavanna. That Noldorin name was changed to be the Sindarin name. In the revision of the Quenta Silmarillion it is said the name Galathilion was a name of Telperion but it was also given to an image made for the Noldor but now with the attachment -the Less.
Then, Galathilion became once again the White Tree of Valinor "The Light of Valinor" which only can be Telperion in an addition to the texts of the white tree in Númenor. But evidently and finally to revisions of the Tree of Númenor it became thus " And a seedling they brought of the White Tree that grew in the midst of Eressëa, and was in turn a seedling of the Tree of Tuna, Galathilion, that Yavanna gave to the Eldar in the Land of the Gods to be a memorial of Telperion, Light of Valinor"
Celeborn.
From Galathilion a seedling was brought to Tol Eressëa and there the seedling grew and became the tree Celeborn. It was planted in the midst of Tol Eressëa. It is not told when the seedling that became Celeborn was brought to Eressëa but obviously it can not have been before Galathilion was given to the Elves upon Túna in Valian Years of the Trees 1142 as it was a seedling if that. Neither is there a telling of how it came to Eressëa.
Basically I see ( purely suggestions ) two major events where a seedling of Galathilion could very well have been brought to Eressëa. The first event could have been when the Teleri at Tol-Eressëa began to long for the light of Aman and for seeing their Elvish friends and Ossë, though he did not liked it, thought them the craft of ship-building and he brought them as parting gift strong-winged white swans to drag the ships and they made their home at Alqualondë. Well arrived the Teleri who forsook Eressëa and moved to Alqualondë may have returned a seedling. If that is the case it would be after the Valian Years of the Trees 1161 but before the end of the First age.
The other major event was after the War of Wrath ending the First Age. For as the war ended all the Elves that fought on Eonwë`s side was permitted to return to the Blessed Land but most of the Noldor that returned rehabited the Lonely Isle, Eressëa, that looks both west and east and that land became very fair, and the Teleri forgave their ancient grief of the Noldor. This may well have been a good time and occasion to bring a fruit of Galathilion to Eressëa.
Nimloth the Less.
In the beginning of the Second age the Elves of Tol Eressëa brought to Númenor, as gifts for the Númenorian people, birds and flowers and herbs and a seedling of Celeborn, the White Tree that grew in the midst of Eressëa. This was planted in the court of the kings in Armenelos and it grew and blossomed and it was called Nimloth the Fair. In the evening and the shadows of night it filled with its fragrance. But near the end of the Second age Sauron came to Númenor and caused it`s fall in year 3319 but a fruit of Nimloth was saved from the fall of Númenor by Elendil and his sons and brought to Middleearth. where it was planted in Minas Ithil.
Minas Ithil`s White Tree.
The sipling was planted in Gondor where it stood in Isildur`s court in Minas Ithil until the city was conquered and the White Tree burned by Sauron in the Second age year 3429. But Isildur had saved a seedling and that was planted in Minas Anor ( Became Minas Tirith ) after the War of the Last Alliance.
Minas Tirith`s first White Tree.
The tree was planted by Isildur and grew in the Court of the Fountain in the citadel of the city. But when the Great Plague devastates Gondor in the Third age year 1636 the Tree dies but a seedling is saved.
Minas Tirith`s second White Tree.
In Third age 1640 the saved seedling is planted the same place in what now had become the King`s House as Osgiliath falls into ruins. Later was build Ecthelion`s Tower and thus the Tree stood beneath the tower. In the later times it does not thrive and in Third age year 2852 it finally dies but is left standing in the citadel.
The new White Tree of Minas Tirith.
There was saved a sipling or a fruit which was the White Tree Gandalf lead Aragorn to find in the mountains at the end of Third age. The White Tree Aragorn II found was not yet seven years old but the fruit which it came from was old and had been lying in the ancient hidden hallow for a very long time.
When the fruit was placed out there, and by whom, can not be told but Gandalf says " before the kings failed or the Tree withered in the court, a fruit must have been set here" So at least it was before Third age year 2852. That statement also tells it was a scion of the Tree in the court and not an even older fruit from perhaps Nimloth.
Who placed it out there ?. That is not said but again some ideas may be followed. And again it is Gandalf who set us on the track. He says to Aragorn "The life within ( the fruit ) may lie sleeping through many long years, and none can foretell the time in which it will awake. Remember this. For if ever a fruit ripens, it should be planted. Here it has lain hidden on the mountain, even as the race of Elendil lay hidden in the wastes of the North."
It may mean nothing and it is my speculations but the words "Before the kings failed" [admitted, taken out of the context ] and his words "Remember this" Could this mean it was only knowledge for the Kings and not known, meant to be known, within the Stewards ? Why had no Steward after 2852 been looking for a Tree or fruit if it was known ? How come the Tree is not older than seven years ? Was the Trees ( and fruits of them ) by some divine interception bound to the line of kings ? meaning they will only sprout, grow and continue to live as long as a King is present or is about to come forth again ?
Gandalf`s words "Even as the race of Elendil lay hidden in the wastes of the North" could imply it was so.