An alternative way would be using the dcast( ) function within the data.table package.
This allows you to convert data from long to wide. In this case, I've created an ID to could the number of records per Tree.
In the re-shaped data, Tree becomes our primary column and circumference is recorded in 7 unique columns (one for each age).
library(data.table)
Orange <- data.table(Orange)[,ID := seq(1:.N), by=Tree]
Orange2 <- dcast(
data = Orange,
formula = Tree ~ ID,
value.var = "circumference")
Orange2
Tree 1 2 3 4 5 6 7
1: 3 30 51 75 108 115 139 140
2: 1 30 58 87 115 120 142 145
3: 5 30 49 81 125 142 174 177
4: 2 33 69 111 156 172 203 203
5: 4 32 62 112 167 179 209 214
EDIT (in response to additional comments/questions):
Technically the data is already ordered by Tree (defined within the data). This is because the variable Tree is a factor variable with preset levels. To order numerically, here are 2 things: (1) Order by as.character( ) and (2) Re-level the variable.
Orange2[order(as.character(Tree),]
1: 1 30 58 87 115 120 142 145
2: 2 33 69 111 156 172 203 203
3: 3 30 51 75 108 115 139 140
4: 4 32 62 112 167 179 209 214
5: 5 30 49 81 125 142 174 177
class(Orange$Tree)
[1] "ordered" "factor"
levels(Orange$Tree)
[1] "3" "1" "5" "2" "4"
Orange2[,Tree := factor(Tree, c("1","2","3","4","5"), ordered = FALSE)]
Orange2[order(Tree),]
Tree 1 2 3 4 5 6 7
1: 1 30 58 87 115 120 142 145
2: 2 33 69 111 156 172 203 203
3: 3 30 51 75 108 115 139 140
4: 4 32 62 112 167 179 209 214
5: 5 30 49 81 125 142 174 177